A318895 -id:A318895 - cp's OEIS frontend

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 2, 2, 1, 7, 1, 2, 2, 2, 1, 4, 1, 8, 1, 2, 1, 7, 1, 2, 2, 5, 1, 6, 1, 2, 1, 2, 1, 14, 1, 4, 1, 3, 1, 11, 2, 5, 2, 2, 1, 9, 1, 2, 2, 27, 1, 4, 1, 3, 1, 4, 1, 20, 1, 2, 2, 2, 1, 6, 1, 11, 3, 2, 1, 9, 1, 2, 1, 4, 1, 8

Offset: 1

Miles Englezou, Jan 13 2025

nonn

Isoclinism is an equivalence relation on groups which generalizes isomorphism: it partitions nonisomorphic groups of the same order into classes. For example, all abelian groups of order k are isoclinic, and therefore belong to a single isoclinism class.
Two groups G and H are isoclinic if: there exists an isomorphism f between the inner automorphism groups Inn(G) and Inn(H); there exists an isomorphism g between the commutator subgroups [G,G] and [H,H]; and if f and g commute with the commutator maps w1:Inn(G)xInn(G) -> [G,G] and w2:Inn(H)xInn(H) -> [H,H].
A diagram of the mappings:
                fxf
Inn(G)xInn(G) ------> Inn(H)xInn(H)
      |                     |
   w1 |                     | w2
      |                     |
      \/                   \/
    [G,G]    -------->    [H,H]
                 g
If the diagram commutes, then G and H are isoclinic.

			a(4) = 1 since both groups of order 4 are abelian and therefore form a single isoclinism class.
a(8) = 2 since of the 5 groups of order 8, 3 are abelian and form a single isoclinism class, and the remaining 2 are isoclinic to each other. Therefore there are 2 isoclinism classes of order 8.

Miles Englezou, GAP Program
The Group Properties Wiki, Isoclinism of groups
Wikipedia, Isoclinism of groups

Cf. A318895, A051532.

A241276 is a lower bound.

GAP
```
# See Miles Englezou link.
```

a(A051532(n)) = 1.

cp's OEIS Frontend

A380147 Number of isoclinism classes of groups of order n.

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